# Riemann theta functions

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##### 1: 21.2 Definitions
###### §21.2(i) RiemannThetaFunctions
For numerical purposes we use the scaled Riemann theta function $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, defined by (Deconinck et al. (2004)), …Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. …
##### 2: 21.8 Abelian Functions
###### §21.8 Abelian Functions
For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …
##### 3: 21.10 Methods of Computation
###### §21.10(ii) RiemannThetaFunctions Associated with a Riemann Surface
• Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

• ##### 4: 21.9 Integrable Equations
###### §21.9 Integrable Equations
Typical examples of such equations are the Korteweg–de Vries equation … Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3). … Magnify
##### 5: 21.1 Special Notation
Uppercase boldface letters are $g\times g$ real or complex matrices. The main functions treated in this chapter are the Riemann theta functions $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, and the Riemann theta functions with characteristics $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$. The function $\Theta(\boldsymbol{{\phi}}|\mathbf{B})=\theta\left(\boldsymbol{{\phi}}/(2\pi i% )\middle|\mathbf{B}/(2\pi i)\right)$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
##### 6: 21.3 Symmetry and Quasi-Periodicity
###### §21.3(ii) RiemannThetaFunctions with Characteristics
…For Riemann theta functions with half-period characteristics, …
##### 7: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
Such a solution is given in terms of a Riemann theta function with two phases. …
##### 8: 21.4 Graphics
###### §21.4 Graphics
Figure 21.4.1 provides surfaces of the scaled Riemann theta function $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, with … For the scaled Riemann theta functions depicted in Figures 21.4.221.4.5 Figure 21.4.4: A real-valued scaled Riemann theta function: θ ^ ⁡ ( i ⁢ x , i ⁢ y | Ω 1 ) , 0 ≤ x ≤ 4 , 0 ≤ y ≤ 4 . … Magnify 3D Help Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: ℜ ⁡ θ ^ ⁡ ( x + i ⁢ y , 0 , 0 | Ω 2 ) , 0 ≤ x ≤ 1 , 0 ≤ y ≤ 3 . … Magnify 3D Help
##### 9: 21.5 Modular Transformations
###### §21.5(i) RiemannThetaFunctions
Equation (21.5.4) is the modular transformation property for Riemann theta functions. The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi(\boldsymbol{{\Gamma}})$ is determinate: …
###### §21.5(ii) RiemannThetaFunctions with Characteristics
For explicit results in the case $g=1$, see §20.7(viii).
##### 10: 21.7 Riemann Surfaces
###### §21.7(i) Connection of RiemannThetaFunctions to Riemann Surfaces
In almost all applications, a Riemann theta function is associated with a compact Riemann surface. … is a Riemann matrix and it is used to define the corresponding Riemann theta function. …